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Physics 42200
Waves & Oscillations
Spring 2013 SemesterMatthew Jones
Lecture 38 – Diffraction
Diffraction
“Light transmitted or diffused, not
only directly, refracted, and reflected,
but also in some other way in the
fourth, breaking.”
Huygens-Fresnel Principle
• Huygens:
– Every point on a wave front acts as a point source
of secondary spherical waves that have the same
phase as the original wave at that point.
• Fresnel:
– The amplitude of the optical field at any point in
the direction of propagation is the superposition
of all wavelets, considering their amplitudes and
relative phases.
Single Slit DiffractionExamples with water waves
• Wide slit: waves are
unaffected
• Narrow slit: source of
spherical waves
• In between: multiple
interfering point sources
Single Slit Diffraction
Think of the slit as a number of point sources
with equal amplitude. Divide the slit into two
pieces and think of the interference between
light in the upper half and light in the lower
half.
Destructive interference when
�2 sin � = �
2Minima when
sin � = � �
12 �
Single Slit Diffraction
sin � ≈ tan � = �/�Minima located at
� = ���� ,� = 1,2,3, …
In general, the “width” of
the image on the screen
is not even close to �.
�
Single Slit Diffraction
Waves from all points in
the slit travel the same
distance to reach the
center and are in phase:
constructive
interference.
Minima located at
sin � = ���� ,� = 1,2,3,…
Minima only occur
when � > �.
Assumptions about the wave
front that impinges on the slit:
• When it’s a plane, the phase
varies linearly across the slit:
Fraunhofer diffraction
• When the phase of the wave
front has significant curvature:
Fresnel diffraction
Fresnel and Fraunhofer DiffractionF
rau
nh
ofe
rF
resn
ell
Fresnel and Fraunhofer Diffraction
• Fraunhofer diffraction
– Far field: � ≫ ��/�
– � is the smaller of the
distance to the source or
to the screen
• Fresnel Diffraction:
– Near field: wave front is
not a plane at the
aperture
�
b
�
�
Single-Slit Fraunhofer Diffraction
Light with intensity �� impinges on a slit
with width �Source strength per unit length: ℇ�Electric field at a distance � due to the
length element ��:
�� = ℇ� !" # ���$ � = %� sin �
� �
�
Single-Slit Fraunhofer Diffraction
�� = ℇ� !&# '() *���
Let � = 0 be at the center of the slit.
Integrate from −� 2⁄ to +� 2⁄ :
Total electric field:
� = ℇ�� / !&# '() *��
01/�
21/�= ℇ�
� !(&1 �⁄ ) '() * − 2!(&1 �⁄ ) '() *
5% sin �= ℇ��
�sin 1
2%� sin �12 %� sin �
� �
�
Single-Slit Fraunhofer Diffraction
� = ℇ���
sin 66
where
6 = 12%� sin �
The intensity of the light will be
� � = � 0 sin 66
�
= � 0 sinc� 6� �
Single-Slit Fraunhofer Diffraction
� � = � 0 sin 66
�= � 0 sinc� 6
Minima occur when
6 = �8, � = 91,92,⋯
6 = 12%� sin � = 8�
� sin � = �8� sin � = ��
Single slit: Fraunhofer diffraction
Adding dimension: long narrow slit
Diffraction most prominent in the
narrow direction.
Emerging light has cylindrical symmetry
Rectangular Aperture Fraunhofer Diffraction
Source strength per unit area: ℇ;�� ≈ ℇ� !&#</= !&>?/=���@
�� = ℇ�
� / !&#</=��01/�
21/�/ !&>?/=�@
0A/�
2A/�
� B, C = � 0 sin6 ′6′
� sin E ′E′
�
Rectangular Aperture
� B, C = � 0 sin 6 ′6′
� sin E ′E′
�
6F = 12%�B/�
EF = 12%�C/�
Double-Slit Fraunhofer Diffraction
• Same idea, but this time we
integrate over two slits:
� = ℇ�� / !&# '() *��
01/�
21/�+ ℇ�
� / !&# '() *��A01/�
A21/�= ℇ��
�sin 66 1 + !&A G!H *
Double-Slit Fraunhofer Diffraction
� = ℇ���
sin 66 1 + !&A G!H *
Light intensity:
� � = 2� 0 sin 66
�1 + cos %� sin �
= 4� 0 sin 66
�cos� E
Where
E = 12%� sin �
Since � > �, cos E oscillates more rapidly
than sin 6
Double Slit: Fraunhofer Diffraction
( ) αβ
βθ 2
2
0 cossin
4
= II( ) θβ sin2/kb≡( ) θα sin2/ka≡
Minima: α = ±π/2, ±3π/2
or: β=mπ, where m=±1, ±2…
( )λθ 2/1sin += ma
ab
λθ mb =sinm=0, ±1, ±2…